• Mar 9th 2010, 04:47 PM
spred
In quadrilateral ABCD, diagonals AC and BD intersect at P. E, F, G, H are the centroids of triangles ABP, BCP, CDP, and ADP respectively. If the area of the quad ABCD is 18, find the area of the quad EFGH. The centroid of a triangle is the point of intersection of the medians of the triangle.

Thank You
• Mar 9th 2010, 11:31 PM
earboth
Quote:

Originally Posted by spred
In quadrilateral ABCD, diagonals AC and BD intersect at P. E, F, G, H are the centroids of triangles ABP, BCP, CDP, and ADP respectively. If the area of the quad ABCD is 18, find the area of the quad EFGH. The centroid of a triangle is the point of intersection of the medians of the triangle.

Thank You

1. Draw a sketch of the quadrilateral. (The difficulty here is to find a quadrilateral which has no further properties)

2. You can prove that

$\displaystyle \overline{HE}\ \parallel\ \overline{GF}\ \parallel\ \overline{DB}$

and

$\displaystyle \overline{EF}\ \parallel\ \overline{HG}\ \parallel\ \overline{AC}$

That means the quadrilateral EFGH must be a parallelogram.

3. The quadrilateral EFGH consists of 4 smaller parallelograms which are part of the 4 triangles which form the quadrilateral ABCD. As an example I've taken the triangle BCP and the corresponding parallelogram KFLP. Use proportions to show that the area of KFLP is $\displaystyle \frac29$ of the triangle BCP.

4. You'll get this result in each of the 4 triangles. Therefore the area of the parallelogram EFGH must be $\displaystyle \frac29$ of the area of the quadrilateral ABCD.